At the most fundamental, no there aren’t that many. But isn’t it considered that any thing that is built up out of bosons is itself a boson? That would mean all the kinds of atoms that exist in nature as well as smaller things.
Whether a particle is a boson or fermion is simply a description of whether the state vector of a system composed of two such particles is symmetric or anti-symmetric when those particles are exchanged. So systems of multiple bosons remain bosons, as you say. But systems composed of multiple fermions can be bosons. The neutrons and protons in helium (
4He) form a bosonic atom even though the particles individually are fermions (essentially, two fermionic systems, and just as -/- and +/+ = +, and -/+, +/- = - in algebraic multiplications, systems of even numbers of fermions can be bosons, and systems of mixed odd numbers of fermions and bosons can be fermions).
But fermions and bosons are not fundamental particles. This is just a description of the symmetry and statistical laws they obey. The particles we have are the quarks, leptons, and gauge bosons: photons, gluons, gravitons, W/Z particles. Combinations of quarks and combinations of leptons can be either bosons or fermions. The gauge particles -- which mediate energy or "force" are all bosons.